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I'm strangely stuck on that one. If I take a set of a finite number of numbers (real numbers, let's be simple), would the average value of rounding/flooring ("take the before-dot part", ie floor(3.9) = 3) each number the same as rounding/flooring the average?

I don't see simple (nor even complex actually) proofs for/against this statement.

Would the same be for more abstract cases, like complex numbers, or infinite sets of elements?

Xenos
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  • Did you try anything ? About any numerical example you will take disproves it. –  Jun 02 '17 at 09:13
  • I actually messed up when doing numerical tries (because it involved dates and months, and I messed up in these); sorry for the feeling of loosing your time :/ – Xenos Jun 02 '17 at 09:34
  • Just remember that the average of a function is never the function of the average (except for the trivial $x\to ax$.) A few cases are useful: the average of logarithms is the logarithm of the geometric mean. The average of inverses is the inverse of the harmonic mean. –  Jun 02 '17 at 09:48

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No. The rounding/flooring is always an integer while the average is rational.

$$\left\lfloor\frac{0+1}2\right\rfloor\ne\frac{\lfloor0\rfloor+\lfloor1\rfloor}2.$$

This property only holds for linear transforms.